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Biology Articles » Biophysics » Medical Biophysics » Memory in receptor–ligand-mediated cell adhesion » Results

Results

The micropipette adhesion frequency assay repeats, sequentially,^{}n tests with a single pair of cells (4). Each test is performed^{}by using computer-automated and piezoelectric translator-driven^{}micromanipulation to control the contact time and area, ensuring^{}it to be as nearly identical to any other tests in the same^{}sequence as possible. Each test generates a random binary adhesion^{}score. The probability of adhesion depends on the kinetic rates^{}of receptor–ligand interaction, surface densities of interacting^{}molecules, and contact time and area.^{}

The result of such n repeated tests is a random sequence whose^{}value X_{i} at the ith position is either 0 or 1. In a previous^{}analysis (4), the running adhesion frequency, defined as F_{i}^{}= (X_{1} + X_{2} + ... + X_{i})/i (1 i n), was plotted vs. i, the test^{}cycle index (Fig. 2A–C). F_{i} fluctuates when i is small^{}because of the small number of statistics, but it should approach^{}an asymptotic curve as i approaches n, unless the sequence is^{}too short. For sequences of sufficient length, F_{n} is the average^{}adhesion frequency, which is the best estimator for the probability^{}of adhesion in each test if the i.i.d. assumption holds, i.e.,^{}the sequence is Bernoullian. A gradual decline in running adhesion^{}frequency, observed for some receptor–ligand interactions^{}(4), could result from receptor extraction from the cell membrane^{}when the two cells are forced to detach from each other. For^{}the three molecular systems studied here, F_{i} approaches a plateau^{}equal to the averaged adhesion probability P_{a}.^{}

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Another way to visualize the sequences in Fig. 2 A–C is^{}to plot the nonzero X_{i}F_{n} vs. i (Fig. 2 D–F). The zero^{}scores for no-adhesion events have been omitted for clarity.^{}Three molecular systems are shown: LFA-1/ICAM-1 interaction^{}(Fig. 2D), TCR/pMHC interaction (Fig. 2E), and homotypic interaction^{}between C-cadherins (Fig. 2F). For each interaction, three sequences^{}were obtained by using three pairs of cells tested under the^{}same conditions. Symbols in Fig. 2 D–F match those in^{}Fig. 2 A–C. The variations in the horizontal levels among^{}the three sequences (i.e., F_{n} values) reflect cell-to-cell variations^{}in heterogeneous cell populations that expressed lognormally^{}distributed receptor–ligand densities. To compare different^{}receptor–ligand interactions, we chose data sets having^{}similar mean adhesion levels (solid lines, Fig. 2 D–F).^{}

Three distinct behaviors seem apparent, even with a brief glance^{}at the adhesion score sequences. Compared with those for the^{}LFA-1/ICAM-1 interaction (Fig. 2D), the sequences for the TCR/pMHC^{}interaction (Fig. 2E) appear more "clustered," whereas those^{}for the C-cadherin interaction (Fig. 2F) are less "clustered."^{}Here, "cluster" refers to consecutive adhesion events uninterrupted^{}by no-adhesion events. In Fig. 2 A–C, clustering manifests^{}as uninterrupted ascending segments in the running adhesion^{}frequency curves.^{}

Fig. 2 suggests that the likelihood of an adhesion in the future^{}test may be influenced by the outcomes of past tests, depending^{}on the biological systems. To analyze this possibility quantitatively,^{}we assume that the adhesion score sequence is Markovian and^{}stationary. The one-step transitional probabilities are independent^{}of the test cycle index i and could be defined in terms of the^{}conditional probabilities:

where n_{ij} is the number of i j transitions^{}directly calculated from the adhesion score sequence; p_{10} +^{}p_{11} = 1 and p_{00} + p_{01} = 1. By definition, p_{01} is the probability^{}of having adhesion (i.e., X_{i+1} = 1) under the condition that^{}the previous adhesion test is not successful (i.e., X_{i} = 0).^{}It is also given a short notation p by definition. p_{11} is the^{}probability of having adhesion in the next test if the previous^{}test also results in adhesion. It is also given a short notation^{}p + p by definition. p represents an increment (positive or^{}negative) in the probability of adhesion in the (i + 1)th test^{}due to the occurrence of adhesion in the ith test, which can^{}be thought of as a memory index. If the i.i.d. assumption holds,^{}p_{01} = p_{11} = p and p = 0, and we recover the Bernoulli sequence.^{}p can be estimated from the average adhesion frequency because^{}it also equals P_{a}. If the i.i.d. assumption is violated (p_{01}^{}p_{11} and p 0), p will not be equal to P_{a} (see Eq. 5 below).^{}

Direct calculations of p_{01} and p_{11} for the data in Fig. 2 D–F^{}show that their values are close to each other and to the averaged^{}adhesion probability, P_{a}, for LFA-1/ICAM-1 interaction, providing^{}preliminary validation for the i.i.d. assumption. By comparison,^{}transition 1 1 is more favorable than transition 0 1 for the^{}TCR/pMHC interaction, whereas the opposite seems true for the^{}C-cadherin interaction, indicating that the i.i.d. assumption^{}might be violated for these two cases.^{}

Closer inspection of the scaled adhesion scores in Fig. 2 D–F^{}reveals that they are clustered at different sizes. It seems^{}intuitive that, for a given "cluster size" m (i.e., m consecutive^{}adhesions), the number of times it appears in an adhesion score^{}sequence contains statistical information about that sequence.^{}Our intuition is supported by the data in Fig. 3, which shows^{}the cluster size distribution enumerated from the adhesion score^{}sequences in Fig. 2. Compared with the distribution for the^{}LFA-1/ICAM-1 interaction (Fig. 3A), the distribution for the^{}TCR/pMHC interaction (Fig. 3B) has more clusters of large size,^{}whereas the distribution for the C-cadherin interaction (Fig. 3C)^{}has more single adhesion events surrounded by no-adhesion events.^{}

To quantify the differences among the three cases in Fig. 3,^{}we derived a formula to express the number, M_{B}, of clusters^{}of size m expected in a Bernoulli sequence of length n and probability^{}p for the positive outcome in each test:

The first term in the upper branch on the right-hand^{}side of Eq. 2 represents summation over the probabilities of^{}having a cluster of size m in all possible positions, i.e.,^{}for clusters starting at i = 2 to i = n – (m – 1).^{}Clusters starting from X_{1} or ending with X_{n} are accounted for^{}by the second term in the upper branch. The lower branch of^{}our formula accounts for the sequence of all 1s. It becomes^{}apparent from the above derivation that Eq. 2 assumes equal^{}probability for the cluster to take any position in the sequence.^{}This can be thought of as a stationary assumption.^{}

The total number of positive adhesion scores in the entire sequence^{}can be calculated by multiplying Eq. 2 by m and then summing^{}over m from 1 to n. It can be shown by direct calculation that^{}mM_{B}(m, n, p) = np. Here, np is the expected total number of^{}adhesion events. This outcome is predicted and shows that Eq. 2^{}is self-consistent.^{}

M_{B} is plotted vs. P_{a} (= p) in Fig. 4A for n = 50 and for clusters^{}of sizes m = 1–4. The actual cluster size distributions^{}enumerated from measured adhesion score sequences (e.g., Fig. 3A)^{}should be realizations of the underlying stochastic process.^{}We used computer simulations to characterize the statistical^{}properties of this stochastic process. The mean ± SEM^{}of the number of clusters of size 1 from 3 (open triangles,^{}mimic experiments where three to five pairs of cells were usually^{}tested) and 50 (filled triangles, a good approximation to Eq. 2)^{}simulated Bernoulli sequences for several P_{a} values are shown^{}in Fig. 4A, which evidently agree well with M_{B}(1, 50, P_{a}) given^{}by Eq. 2.^{}

We next extend Eq. 2 to the case of a Markov sequence by including^{}a single-step memory. The four conditional probabilities defined^{}in Eq. 1 form a one-step transition probability matrix [P] of^{}a stationary Markov sequence. Using Bayes' theorem for total^{}probability, the unconditional probabilities for the (i + 1)th^{}test are related to those for the ith test by [P]:

By applying the Chapman–Kolmogorov equation^{}(8), the n-step transition matrix can be obtained as [P^{(n)}]^{}= [P]^{n}. With the initial condition of P(X_{1} = 1) = p and P(X_{1}^{}= 0) = 1 – p, it can be shown by mathematical induction^{}that P(X_{i} = 1) = p(1 – p^{i})/(1 – p). The formula^{}for the expected cluster size distribution can now be extended^{}as follows:

Setting p = 0 reduces Eq. 4 to Eq. 2, as required. Eq. 4 has^{}another special case at p = 1 – p when it simplifies to^{}M_{M}(m, n, p, 1 – p) = p(1 – p)^{n}^{–m}, which describes^{}the situation where the first adhesion event in the sequence^{}would increase the probability for adhesion in the next test^{}to 1, leading to continuous adhesion for all tests until the^{}end of the experiment.^{}

Experimentally, the adhesion probability can be estimated from^{}the adhesion frequency F_{n}. The expected value of F_{n} can be calculated^{}as follows:

If p = 0 (i.e., a Bernoulli sequence), F_{n} approaches p as n^{}becomes large. However, if p 0 (i.e., a Markov sequence), then^{}F_{n} approaches p/(1 – p).^{}

M_{M} is plotted vs. P_{a} (related to p and p by Eq. 5) in Fig. 4B^{}for n = 50, m = 1 (i.e., solitary 1s bound by 0s from both ends)^{}for p ranging from –0.3 to 0.5 in increments of 0.1. The^{}case of p = 0 (i.e., Bernoulli sequence) is plotted as a solid^{}curve. Cases of p > 0 (i.e., memory with positive feedback)^{}are plotted as dotted curves. Cases of p < 0 (i.e., memory^{}with negative feedback) are plotted as dashed curves. Increasing^{}p shifts the curve downward toward a smaller number of isolated^{}1s in an adhesion score sequence.^{}

It can be shown by direct calculation that mM_{M}(m, n, p, p) = p[n – p(1 – p^{n})/(1^{}– p)]/(1 – p). Here, the left-hand side sums adhesion^{}events distributed in various clusters of different sizes, and^{}the right-hand side is the expected number of adhesion events^{}in n repeated tests, nE(F_{n}) (cf. Eq. 5). This predicted result^{}confirms that Eq. 4 is self-consistent.^{}

In Fig. 3, Eq. 4 was fit (curves) to the measured cluster size^{}distributions (bars) (see Materials and Methods for procedure^{}detail). Three different types of behaviors can be clearly discerned.^{}A memory index p 0 was returned from fitting the LFA-1/ICAM-1^{}data in Fig. 3A. Because of the limited amount of data, small^{}fluctuations of p from zero could be observed even for Bernoulli^{}sequences, as seen from computer simulations of a small number^{}of cell pairs. Fitting the TCR/pMHC data (Fig. 3B) returned^{}a positive p, indicating the presence of memory with positive^{}feedback, whereas fitting the C-cadherin data (Fig. 3C) returned^{}a negative p, indicating the presence of memory with negative^{}feedback. These results support our preliminary conclusions^{}based on the observations in Fig. 2 and preliminary analysis^{}using the p_{01}, p_{11}, and P_{a} comparison.^{}

The above fittings used the distribution of clusters of all^{}sizes (i.e., all m values) for a given P_{a} to evaluate p. However,^{}differences in the distribution of cluster sizes are dominated^{}by the difference in the expected number of clusters of size^{}1 (Fig. 3, comparing the first bar in each panel). Thus, the^{}number of clusters of size 1 in experimental adhesion score^{}sequences can be used as a simple indicator to evaluate the^{}memory effect.^{}

Analysis so far has used the raw data shown in Fig. 2, which^{}have similar P_{a} values. To obtain further support with 10x more^{}data, we varied contact times and/or receptor–ligand densities^{}to obtain different average adhesion frequencies, which also^{}allowed us to examine the potential effects of molecular densities^{}on p through P_{a}. For each system, the experimental number of^{}clusters of size 1, M_{exp}(1), for each P_{a} value was plotted in^{}Fig. 4B to compare with the theoretical curves, M_{M}[1, 50, p(50,^{}P_{a}, p), p]. It can be seen that the LFA-1/ICAM-1 data are scattered^{}evenly from both sides of the solid curve corresponding to p^{}= 0. By comparison, almost all of the TCR/pMHC data are below^{}the theoretical curve for p = 0, and most of the C-cadherin^{}data are above that curve. These results further support our^{}conclusions regarding three types of behaviors.^{}

The memory index p was obtained from fitting experimental cluster^{}size distribution with Eq. 4 and is plotted in Fig. 5(solid^{}bars) along with p estimated from direct calculation (using^{}Eq. 6 below) (open bars). Comparable results were obtained by^{}both approaches for all P_{a} values tested for all three systems^{}exhibiting qualitatively distinct behaviors. p values for the^{}LFA-1/ICAM-1 system are not statistically significantly different^{}from zero (P value 0.23) except in one instance (P value =^{}0.06). By comparison, a vast majority of the p values for the^{}TCR/pMHC system are statistically significantly greater than^{}zero, whereas half of the p values for the C-cadherin system^{}are statistically significantly less than zero and are marked^{}with asterisks on the top of corresponding solid bars to indicate^{}P value 0.05.^{}

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